Question

There are fuo places \(\mathrm{X}\) and \(Y, 200 \mathrm{~km}\) apart from each other. Initially two persons \(P\) and \(Q\) both are at \(^{\circ} X:\) The speed of \(P\) is \(20 \mathrm{~km} / \mathrm{h}\) and speed of \(Q\) is \(30 \mathrm{~km} / \mathrm{h}\). Later on they starts to move to and fro benveen \(X\) and \(Y\). If they starts to move between \(X\) and \(Y\), then for the first time when they will meet each other? (a) after 12 hours (b) after 24 hours (c) after 30 hours (d) after 8 hours

Short answer

Answer: P and Q will first meet after 2 hours.

Step-by-step solution

Calculate the total relative speed

To find their total relative speed, we'll add the speeds of P and Q since they are moving in the same direction. That gives us \(20 \mathrm{~km/h} + 30 \mathrm{~km/h} = 50 \mathrm{~km/h}\).

Calculate the time for both to travel 200 km

Divide the distance, 200 km, by the total relative speed, 50 km/h: \(\frac{200 \mathrm{~km}}{50 \mathrm{~km/h}} = 4 \mathrm{~hours}\). It will take them 4 hours to travel the distance of 200 km.

Calculate the time they first meet

Since both are moving back and forth between X and Y, they will meet at the halfway point as they travel in the same direction. This point is 100 km away. To find the time it will take for them to meet, use the formula: time = total distance / relative speed. In this case, the total distance is 100 km and the relative speed is 50 km/h: \(\frac{100 \mathrm{~km}}{50 \mathrm{~km/h}}=2 \mathrm{~hours}\).

Conclusion

Thus, for the first time, they will meet each other after 2 hours. The correct answer is none of the given options.

Key concepts

Speed and Distance

Understanding the concepts of speed and distance is crucial in solving many physics and math problems. Speed defines how fast an object is moving, and it is calculated by the formula:
\[\text{Speed} = \frac{\text{Distance}}{\text{Time}}\]
Distance, on the other hand, is the total path covered by a moving object. In our scenario, we have two individuals, P and Q, starting from the same point X and heading towards point Y, which is 200 km away.

• Person P moves at a speed of 20 km/h.
• Person Q moves at a speed of 30 km/h.

Since both are moving between points X and Y, their movements contribute to determining when they first meet after departing in different directions.

Time Calculation

Time calculation is an essential part of determining when two moving objects will meet. It requires knowing speeds and distances to accurately compute the time it takes for events to occur. When dealing with two objects moving towards one another, calculating their relative speed helps in finding the time until they meet.
In our problem:

• Total distance between X and Y is 200 km.
• Relative speed when moving in the same direction is the sum of both speeds—20 km/h (P) + 30 km/h (Q), totaling 50 km/h.
• Time is determined by dividing the total distance by this relative speed.

This calculation simplifies the prediction of when exactly they will meet on their round trips.

Problem Solving Steps

Solving problems involving speed, distance, and time involves a systematic approach to ensure accurate results. Here's a breakdown of the steps we used:

• Calculate the relative speed. For bodies moving in the same direction, simply add their speeds.
• Determine how long it would take to cover the full distance between their start points—200 km in our problem.
• Because they meet halfway through their first cycle of movement, compute half of that distance as their meeting point.
• Using the meeting point distance (100 km), determine the time using the calculated relative speed.

Following these steps carefully leads to the conclusion that they will meet after 2 hours of travel, proving none of the given multiple-choice answers correct.

Distance Calculation

Calculating distances accurately is key when dealing with motion-related problems. Distance becomes particularly interesting when dealing with objects moving back and forth, such as P and Q.

• The distance between X and Y is initially given as 200 km.
• Since they are moving back and forth, focus on the part that makes them meet the first time, halfway at 100 km.

Breaking down problems by moving step-by-step with clear distance references ensures correct solutions. Always align distance calculations with the direction and speed of movement to avoid errors.